DTIC ADA040475: Titan IIIC Guidance with the Carousel VB Inertial Guidance System. Reissue A

REPORT SAMSO- TR-75-297
{SUPERSEDES TR-73-220)

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Titan IIIC Guidance with the Carousel VB
Inertial Guidance System

T. E. DARNAUD and J.B.SHAUL

Guidance and Control Division
Engineering Science Operations
The Aerospace Corporation
El Segundo, Calif. 90245

\

i

15 July 1975

Prepared for

SPACE AND MISSILE SYSTEMS ORGANIZATION
AIR FORCE SYSTEMS COMMAND
Los Angeles Air Force Station
P.0. Box 92960, Worldway Postal Center
Los Angeles, Calif. 90009

Approved

1/

/J. R. Allder, Director
Guidance and Flight Dynamics
Subdivision

Guidance and Control Division
Engineering Science Operations

Cj).

D7 Baxter, Director
Titan III Directorate
Vehicle Systems Division
Systems Engineering Operations

Publication of this report does not constitute Air Force approval of
the report's findings or conclusions. It is published only for the exchange
and stimulation of ideas.

^RICHARD E. WOLFSBERGER, Cornel, USAF
Assistant Program Director
Expendable Launch Vehicles SPO

This report supersedes and replaces TR-0073(34 1 3 -02)- 1 , dated 29
November 1972.

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Space and Missile Systems Organization
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>9. KEY WOROS ( Continue on nvirat «id« If nKMitry and Identity by bloc* nunbar)

Guidance Equations
Inertial Navigation ~

Carousel VB -3

, Rocket Steering Law
\ Vehicle-Borne Computer Program

BSTRACT (Contlnua on ravmraa alda It ttacaaaary and tdantlty by block numb«0

The Titan II^C Standard Space Launch Vehicle, starting with Vehicle 26,
will use the Carousel VB Inertial Guidance System for navigation, guidance,
and digital flight controls. ''This guidance system consists of a Carousel VB
Inertial Measurement Unit (1MU) and a MAGIC 352 Missile Guidance Com-
puter (MGC), both manufactured by Delco Electronics, General Motors
Corporation, The C-VB IMU is a modification of the C-IVB inertial navi-
gator currently in airline service as a primary navigation system. — ^

00 ruKM 1473

irACSiMlLCl

a

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ABSTRACT (Conttnumd)

' A unique feature of the Carousel IMU is that two of the three gyro/
accelerometer sets are revolved at 1 rpm with respect to the stable
platform. The effect is to partially cancel certain instrument errors
associated with the revolving instruments. Conversion of this instrument
from commercial airline navigation service to guidance and control of
the rocket boost vehicle presented a guidance software design task that
is the subject of this paper.

UNCLASSIFIED

SECURITY CLASSIFICATION OF TNI* RAOCfOM* Oat*

PREFACE

The guidance equations derived in this paper were developed by
the authors in a joint effort with D. L. Kleinbub and A. C. Liang of The
Aerospace Corporation.

A summary of the first issue of this report was presented at the
Sixth Hawaii International Conference on System Sciences January 11, 1973
and was published in the proceedings.

CONTENTS

m

:\ m

m

fM

:ni poduction

MISSION SUMMARY AND VEHICLE DESCRIPTION

General

Vehicle Description . . . .

1 . Stage 0

2. Core Stages I and II

Rage III

Control Module

Mission Description (Flight Plan VIU
1 . Ascent to Parking Orbit

2. Parking Orbit

3. Stage HI First Burn

4. Transfer Orbit

5. Final Orbit Injection

6. Paylcad Separation

GUIDANCE, STEERING, AND NAVIGATION

Gene ral

Guidance

Thrust Act deration Prediction

Pitch Stee ring
Yaw Steering

Estimate of Time to Go

5. Integral Control . . . .

6. Maneuvering Equations

Navigation

Accelerometer Resolution and
Compensation

2. Gyro Drift Compensation

f PRECEDING PIGS BLANK-HOT FUME]

' T*

-3-

CONTENTS (Continued)

3. Transformation of Velocity to Inertia]

x, y, 7, Coordinate System

4. Vehicle Inertial Position .

5. Gravity Computations

6. Vehicle Inertial Velocity

D. Attitude Errors .

1 . Synchros

2. Attitude Error Computations

REFERENCES

FIGURES

1. Titan I1IC External Prolile . ... .

2. Typical Titan IIIC Mission Profile

3. Carousel VB IMU Geometry

4 Navigation Block Diagram

TABLE

1 . Carousel VB IGS Characteristics . -

SECTION I

INTRODUCTION

Referent el is a complete specification of the guidance equations
for the Titan HIC launch vehicle. It includes all information necessary for
programming the launch vehicle guidance computer. Reference 1 does not,
however, include any discussion of the guidance algorithms or the derivation
of equations. The equations specified by Ref. 1 and discussed in this
document are applicable to SSLV C-26 and subsequent vehicles.

Section II, a very general description of a typical Flight Plan VII
mission and of the Titan IIIC vehicle, is included as an aid to understanding
the inflight equations, some aspects of which arc vehicle- and/or rnission-
peculia r.

Section III is a discussion of the Titan IIIC navigation, guidance,
and steering.

-5-

SECTION II

MISSION SUMMARY AND VEHICLE DESCRIPTION

A. GENERAL

The Flight Plan VII mission consists of injecting a payload(s) into a
synchronous or near- synchronous equatorial orbit using the Titan IIIC SSLV.
The combined vehicle Stages II and III, together with the payload, are
injected directly into an e'liptical parking orbit having a perigee of approxi-
mately 80 nmi and an apogee of approximately 235 nmi. At the first
descending or ascending node, depending on mission options, a first burn of
Stage III produces an elliptical orbit with an apogee of approximately 19,323
nmi and an orbital inclination . educed by approximately 2-1/4 deg. At apogee,
a second Stage III burn produces the desired synchronous (or near synchronous)
equatorial orbit.

B ■ VEHICLE DESCRIPTION

The Titan IIIC launch vehicle shown in Fig. 1 consists of a solid
motor stage and three liquid engine stages plus a control module. The solid
motor stage is referred to as Stage 0, and the three liquid engine stages are
referred to as Core Stages I c II, and Stage III. The control module and
Stage III are sometimes collecti vely called the transtage. The stages are
briefly^ described in the following paragraphs.

1. STAGE 0

Stage 0 consists of two solid propellant rocket motors positioned
parallel to the standard core in the yaw plane. Thrust vector control is
accomplished by injecting oxidizer (N^O^), pressurized by gaseous nitrogen
(N^), into any of four quadrants of the nozzles.

PRECuDINO PAJS BLANK-HOT PIIirsB" "

-7-

1

PAYLOAD FAIRING-

STATION

FAIRING AND PAYLOAD-
INTERFACE

STAGE il AND STAGE III-
SEPAR ATION PLANE

-CONTROL MODULE

STAGE II

STAGE I A.'O STAGE II-
SEPAR ATION ’’LANE

/'7°h

' l\ ]/ — 24. 00 R

PK\ “ ' - 360- 15

| \ V-,z

125.80

(typ each segment)

STAGE I —

-12.00 R

STAGE 0 (2 SRMs)-

5 SEGMENT SOLID-

6 OFFSET v

r T "1 14-

-TVC TANK

H \

2° OFF SET —

S^O’ GIMBAL-Jf-^n 5° 3 O' GIMBAL p

(max)

—1213. 10
—1224. 3 1
1240.68
— 1274. 12

1304.25

—1326.00

—1348. 10 (nozzle exit)
- 1375.79

ROLL AXIS

HEATSHIELD —r'/ b.L.O.

THRUST CHAMBER—' |
COVERS !

-117° 24'
(typ)

85.00 R-

(typ)

<fc. YAW AXIS^jargex

126. 51

(typ) I

-W. L. 60

-<E PITCH AXIS

Figure 1. Titan IIIC External Profile

-8-

z.

CORE STAGES I AND II

Stages I and II are powered, respectively, by dual and single *hrust
chamber, turbopump-fed, liquid propulsion systems. These systems utilize
storable propellants, a 50:50 mixture of hydrazine and unsymmetrical
dimethylhydrazine (UDMH) for fuel and nitrogen tetroxide (N^O^) for oxidizer.

3. STAGE III

The Stage III propulsion system consists of two pressure-fed engines.
Stage III also contains an attitude control system (ACS), which provides atti-
tude control during coast, ullage control prior to main engine burns, and
velocity additions for vernier phases and for satellite ejections. The pro-
pellants used for the main propulsion system are the same as for Stages I
and II. The attitude control system employs hydrazine monopropellant
engines.

4. CONTROL MODULE

The control module contains the major elements of the flight control,
inertial guidance, electrical, telemetry, tracking, and flight safety systems.
The control module is attached to the forward end of Stage III and remains
attached throughout flight.

C. MISSION DESCRIPTION (FLIGHT PLAN VII)

The following is a brief mission description of a typical Flight Plan
VII. All values given are only approximate, and any discussion of the Flight
Plan that does not contribute to understanding the guidance equations has been
intentionally omitted. A complete description of a Flight Plan VII mission
is given in Refs. 2 and 3. Figure 2 is an illustration of the Flight Plan.

1. ASCENT TO PARKING ORBIT

The vehicle is launched vertically from Air Force Eastern Test
Range (ETR) Pad 41 or 40; shortly after liftoff, it is rolled from a pad azimuth
of 100. 2 deg to a flight azimuth of 93 deg. Following the roll maneuver, an
open-loop pitchover, with load relief during the region of max Q, is performed
for the remainder of Stage 0 burn. Stages I and II are closed-loop guided;

-9-

ss

•H

the engines burn for approximately 149 and 207 sec, respectively, and the
stages burn to propellant exhaustion. At Stage II propellant depletion, the
vehicle is nominally in an elliptical orbit 80 X 235 nmi. Further, in the case
of a low performing vehicle, the minimum orbit is 80 X 95 nmi.

2. PARKING ORBIT

Immediately after injection into the parking orbit, the vehicle attitude
is adjusted to the standard orbital orientation: vehicle longitudinal axis
normal to the geocentric radius vector, the nose of the vehicle pointed approxi-
mately along the velocity vector, and the vehicle pitch plane coincident with
the orbital plane.

After coasting in the parking orbit to near the f-rst descending or
ascending node, the vehicle is reoriented in yaw as a startup attitude for the
first burn of the transtage. Concurrently, a roll maneuver is performed
so that the vehicle telemetry antenna is pointed toward the desired ground
tracking station upon completion of the reorientation maneuver.

3. STAGE III FIRST BURN

Close to the equatorial crossing, the Stage III engines are ignited
and burn for approximately 300 sec. The vehicle is injected into an elliptical
orbit with an apogee of 19, 323 nmi. The first burn includes an orbital plane
change maneuver of approximately 2. 25 deg. If required, the ACS of
Stage III can adjust the orbital parameters by a vernier velocity addition.
Nominally, shutdown of the Stage III engines is controlled so that a 6- sec
ACS vernier phase is required.

4. TRANSFER ORBIT

During the transfer orbit, the transtage performs certain maneuvers
to meet thermal control requirements. Currently, three options axe being
considered for thermal maneuver. The first, called rotisserie, is an oscilla-
tory roll maneuver of approximately ±115 deg at a rate of 1 deg/sec, with a

-11-

f -min dwell time at cat. h cxlremo position. In the sei ond maneuver,
toasting. th< vehicle is simply turned hack and forth in space at widely-
spaced time intervals, l’he third is a continuous roll maneuver between 1
and 2 deg/ set. In all three options, the relationship of the sun vector to the
vehicle or spacecraft axes is specified by the payload thermal requirements.
In addition to the thermal maneuvers, the vehicle is oriented several times
during the transfer orbit tc an attitude that permits reliable reception of
telemetry.

Finally, shortly before reignition of the Stage III engines, the
vohic le is reoriented to a startup attitude for the second burn, which also
points the telemetry antenna earthward.

n. FINAL CRB IT INJECTION

The second Stage III burn, of approximately 104- sec duration,
injects the payload and /ehicle into a circular orbit at 19,323-nmi attitude,
with a near zero-deg inclination.

6. PAYLOAD SEPARATION

After the second Stage III shutdown, the vehicle is reoriented for
the satellite separation phase. This orientation can vary depending on the
selected mission. After sufficient time for stabilization at the desired atti-
tude, the ACS is switched (o the payload release coast mode; after sufficient
time for stabilization in this mode, the payloaa is released. At this point,
the Titan IIIC mission can be ended with a transtage shutdown sequence.
Alternatively, the equations provide an option for performing a short
propellant settling burn, another payload release sequence, and, finally, the
transtage shutdown sequence.

In both of these payload release options, the equations issue a
variety of required discretes and can perform multiple reorientations as
specified by payload requirements.

-12-

SECTION III

GUIDANCE, STEERING, AND NAVIGATION

A. GENERAL

The vehicleborne digital computer program is divided into three
distinct sections: digital flight control equations, ground equations, and

guidance equations. This document treats only the guidance equations; the
other programs are mentioned only when they interface with the guidance
equations. The carousel sensor and gimbal geometry is given in Fig. 3,
while principal computer and sensor characteristics are tabulated in Table 1.
Figure 3 and Table 1 are modifications of data taken from Ref. 9.

The guidance equations are organized into various subprograms.

A main control or executive program controls logical routing to appropriate
calculations , depending on the phase of flight and/or significant flight events.
The navigation equations accept inertial measurement unit (IMU) data and
compute current vehicle position and velocity. Booster steering equations
calculate open-loop steering for Stage 0. Powered flight equations provide
vehicle steering as a function of navigational quantities and desired end con-
ditions. Several coast phase subprograms provide computations for desired
vehicle attitudes and event initiations during nonpowered flight. Finally, the
specific flight plan and the multistage Titan 1IIC vehicle necessitate numerous
subprograms to repeatedly reinitialize the guidance equations for each phase
of flight and to issue discretes to control events such as engine shutdown,
engine ignition, and stage separation. The following paragraphs describe
the guidance philosophy and overall logic flow of the guidance equations.

B. GUIDANCE

The guidance philosophy governing the guidance equations presented
in this document is commonly termed 'explicit" guidance. Generally, an

-13-

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- 14-

explicit guidance scheme is one in which an explicit solution to the powered
flight dynamics is computed in flight and the desired end result of guidam e
is explicitly defined in the guidance equations. The Flight Plan VII guid-
ance equations, deoenclmg upon the flight phase, define different desired
end conditions. The burns of Core Stages I and II are planned for a specific
position magnitude, velocity magnitude, radial velocity, and preplanned
orbit planes. The end result of guidance for the first burn of Stage III
(transtage) is injection onto a transfer ellipse whose apogee, semi-major
axis orientation, and orbital plane are specified. Finally, during the second
burn, the transtage guides to a near-circular orbit at the transfer ellipse
apogee radius and the orbit piano is specified (approximately equatorial).

References 4, 5, and 6 all contain general discussions of explicit
guidance algorithms.

1. THRUST ACCELERATION PREDICTION

A basic requirement of the guidance equations presented here is a
prediction of vehicle performam c, particularly of thrust acceleration. For
this report, "thrust acceleration" is used as an abbreviation for "accelera-
tion (of the vehicle) due to thrust. " The following paragraphs derive a
time-dependent expression for thrust acceleration.

Consider the variable T , defined as

r

= W/W (1)

whe re

W - total weight of vehicle at any time
W • rate of change of vehicle weight

T^ can be interpreted as the time remaining to zero vehicle weight (mass
intercept), if a constant W is assumed.

-16-

Since specific impulse is

1 = Thrust/ W

sp

and thrust acceleration is

Ath -- Thrust/ (W/g)
by solving for thrust and weight as

(2)

(3)

W = T W

Thrust = W1

sp

(4)

one can express A„,, as
1 H

A—u - gl /T
TH 6 sp r

(5)

Further, since the effective exhaust velocity is given as

V = gl
e 6 sp

(6)

then it follows that

ATH = Ve/Tr

(7)

Finally, since T^ is a linear function of time, then A^,^ as a

function of time t can be expressed as

ATH'1' - Ve/(TrO*t)

(8)

where TrQ = value of Tr computed at t = 0.

-17-

2. PITCH STEERING

I" polar coordinates , the two-body equations of motion for a vehicle
in a central for. c field gives the radial acceleration as:

^ ^THr ”

wher<

^THr ~ rdciial thrust acceleration

L. n

w *'■ centrifugal acceleration
2

|i / R ~ acceleration due to gravity

If a time-dependent guidance steering law for the radial thrust
pointing d1 rection is assumed to be

ATHr/ATH T A ' Bli f WR2 - ci2R)/Ath]
then, upon substituting for A^, one finds that Eq. (9) becomes

RMA-tBt>ATH (11)

This steering law, the familiar "linear sine" law, has been applied
to numerous guidance programs. It has the advantages of simplicity and
very near fuel optimality for current ro''kof-r'-.'>'.vc'nc,i u.

Since explicit guidance equations require a prediction of vehicle
motion, it is desirable to time-integrate Eq. (1 1} twice to estimate future
vehicle radial velocity and position. However, before these integrations
are attempted, Eq. (11) can be simplified to reduce integration complexities.

- 18-

The* linear assumption of T can be written as

t

(12;

Substituting Eq. (12) into Eq. (11) yields

R = (A + BTr0 - BTr)(ATH)

(13:

And, after some algebraic manipulation,

R = -BT Ath + (A + BTrQ)(ATH)

= -BV + (A i BT r) (A_u)
e rO TH

(14)

In Eq. (14), the quantities A,B,Vo, and are assumed constant;

therefore, Eq. (14) can be rewritten as

R = A' + B A

r r T H

(1 5>

Time integration of Eq. (15) from a given time (t = 0) to cutoff (t - T ) yields

Vrr V

rT rT

+ / 8 A' dt + B / 8 A„udt

r Jn rJ 0TH

(16)

/•T ft

f, % *>

R, = R + V T +
f r g

dt + B*fn 8 f ATHdS

JO Jo

dt

-19-

where

&S* ■' s

M 4

WM

|g||

il

V = present radial velocity
r 1

V r - radial velocity at cutoff
rf

iir

%k4.

m

R = present ra fiai position
= radial position at cutoff

The integrals of thrust acceleration in Eq. (16) can be evaluated
using the time-dependent expression for A^^, given in Eq. ( 1 Z) :

nil

h M

r

/ s

Jo

A.dt - V
Th g

A ds di - V T - V T 4 V T
IH g g g rO eg

in these integrals, it is assumed that V is known. A derivation of the
b g

second integral (a^) can be found in practically any discussion of an explicit
guidance technique, such as R .. 4.

To solve for the steering coefficients A^ and Bf> one must substitute
the values for the integrals in Eq. (17) into Eos. (16):

V c - V 4 A' T + B V
rf r r g r g

R, R + V T 4 A' (T2/2) 4 B (a, ,)
I r g r g r 1 2

it

(19)

Solving for yields

B = R, - R - (T / 2)( V , + V )
r f g ” rf r

Tro - <y 2>

+ V T
e g

and for A',
r

A' = (1/T )(V , - V - V B )
r g rf r g r

(20)

Finally, the expression for the desired radial acceleration A^ .s
obtained by substitution into Eq. (15):

A

r

R = (1 /T )(V , - V
g rf r

- V B ) + B A,
g r' r

rH

(21)

3. YAW STEERING

The yaw steering is derived in a manner analogous to the pitch

steering scheme. In the yaw case, a desired orbital plane is specified uy a

vector normal to the plane. Normal error parameters R and can be

computed as the dot product of vehicle inertial position and velocity with the

specified normal vector. Since the desired value of both error parameters

is zero at cutoff, the following equations can be used to determine the yaw

steering coefficients A and B :
f n n

0 = V + A T + B V
n n g n g

0 = R + V T + A T / 2 + B a, _
n n g n g n 1 2

(22)

-21-

4.

ESTIMATE. OF TIME TO C.O

* .1

The estimate of time to go is derived from AM, the vehicle angular
momentum to bo gained, which is given as

AM = Mf - M

"M

where

r<;

. «

M^ = desired vehicle angular momentum magnitude at cutoff
M - present vehicle angular momentum magnitude

Angular momentum to be gained is also given by the integral equation

AM =

where T ^ is the time until rocket engine cutoff.

To estimate time to go, solve the integral Eq. (24) for T by a

8

numerical integration technique. The magnitude of angular momentum is
given as

M = Rv

v^ - velocity in the tangential direction

R = position magnitude

Per unit mass

Differentiating Eq. (25) produces

M = Rvt + Rv^

(26)

Another expression based on orbital and tangential thrust acceleration is

\ = -\R/R + ATHt (27)

where is the vehicle thrust acceleration magnitude in the tangential

direction. Substituting the expression for into Eq. (26) yields for M the
simple expression

M = Rvt + R(-vtR/R + ATHt) = RATHt (28)

The integral of Eq. (24) is evaluated using Simpson's integration

formula, with the integrand computed at t = 0, t = T , and three equally-

6

spaced points in between.

The first factor in the integrand, position magnitude, is computed at
the five desired time points by a linear interpolation between the present
vehicle position magnitude R and the final position magnitude R^; thus,

R(tj) = R + (j - l)(Rf - R)/4 j = i, ...5 (29)

Thrust acceleration magnitude as a function of time has been derived m
Paragraph III. B. 1. Thus, it follows that

ath<V ^

V

J '

Tr0- -

J )T / 4
g

j = 1.

(30)

Where T is tnc time-to-go from the last major cycle decremented by 1 sec

g

-23-

The tangential acceleration A,p^ is obtained from a square root as

, 1/2

ATHt(y =

ATH(V " ATHr{V " ATHn(V

(31)

AIL tliat remains is to compute the normal and radial thrust accelerations

ATIIn anc* ATHr at Rie re<5uire<^ <-irne points.

The present radial components of thrust acceleration have already

been given as

A

THr

a; + BrATH - (-p/R2 + u>2R)

(32)

and at cutoff as

A_ir . - A' +
THrf r

B

-ATHf ” + wf Bf)

(33)

where the subscript f designates final values at cutoff. The centrifugal
acceleration terms in Eqs. (32) and (33) can be rewritten, respectively, as

2

tjJ

R =

(34)

? 2
w*Rf = M{

Finally, by linearly interpolating the accelerations due to gravity and centrifu-
gal force and by employing the value of acceleration magnitude derived pre-
viously in Eq. (30), one obtains for A^,^^ at the required time points

ATHr(tj) = Ar + BrATH(tj} " <^/r2 + m2/r3)

- (j - l){[(-p/R2 + M2/Rf3) - (-p/R2 + M2/R3)]/4|

(35)

1, ... 5

-24-

The normal component of thrust acceleration is given simply as

ATHn(V = An + BnATH(V J = 1 > ■■■ 5 (36)

By combining the results of Eqs. (29) and (31), one obtains the integrand of
Eq. (24) at the specified time points

M(t.) = R(t.)ATHt(t.) (37)

and, using Simpson's integration rule, one can compute an estimate of AM:

AM' = [M(t} ) + 4M(t2) + 2M(t3) + 4M(t4) + M(t5)]T'/l2 (38)

where AM' is the angular momentum if the initial estimate of time to go is
assumed. By comparing AM, the true angular momentum to be gained, with
AM', one can compute an adjustment to the time to go T

it

Ta = (AM - AM')/M(t5) (39)

Finally, the time to go becomes

T

g

= T' + T
g a

(40)

With time to go T and the time to mass depletion T , the acceleration

O 1*

integral V is computed as
S

V

g

Iv /(T
j e ' rO

t) dt = -V ln(l - T IT J
e o' r 0

(41)

-25-

The guidance scheme described here is iterative. The steering coefficient
calculations require an estimate of time to go T , while the time-to-go
calculations presuppose a knowledge of the steering coefficients. Fortu-
nately, experience has shown that with reasonable startup values and
appropriate updating between major guidance cycles, the equations described
here ccnverge quite rapidly for the mission considered.

5. INTEGRAL CONTROL

In Paragraphs III. B. 2 and III. B. 3, expression's were derived for
desired radial and normal thrust accelerations. If these accelerations were
translated directly into vehicle axis pointing-direction commands, a thrust
acceleration pointing e**ror would result because of vehicle ard IMU prop-
erties, such as engine misalignments, vehicle center of gravity effects,
and IMU gimbal misalignments. Integral control, the method for overcoming
these pointing errors, is used to measure differences between desired and
actual guidance- computed accelerations. These differences are numerically
time-integrated with an appropriate weighting factor for stability reasons,
i.e. , a digital filter, and finally subtracted from the desired radial and
normal accelerations. The final steady state effect is to bias the vehicle-
pointing commands so that the observed misalignments are canceled.
Preference 7 contains a complete description of integral control. The
equations are mechanized as

AA,rAAn + (C:F6} - AnJ l4Z)

AAr-AAr + (cF6) [(AVr/At) - Aj (43)

The arrow indicates that the results are stored in the original memory
location .

-26-

where

i

i

i

>

AA and AA
n r

integral control terms

Cj.,^ integral weighting factor
A^ - desired normal component of total vehicle acceleration
A - desired radial component of total vehicle acceleration

AV /At - guidance-computed normal component of total vehicle
acceleration

AV /At guidance - computed radial component of total vehicle
acceleration

At 1 -second major cycle interval

The corrected desired normal and radial equations are then given as

nc

= A

n

kA

(44)

A

rc

2

u> r

(45)

Equation (45) includes terms for gravity and centrifugal accelerations, these
terms were added so that A^, excluding the effect of AAf) would represent
the desired radial thrust acceleration.

Dividing Eqs. (44) and (45) by thrust acceleration magnitude yields
the desired unit vehicle longitudinal axis in the radial, normal, and tangential
coordinates

ID = A /A_„
&n nc TH

(46)

ID = A / A u
&r rc TH

(47)

1/2

U£t * (* - °*n - °fr)

(48)

-27-

Then the unit desired vehicle longitudinal roll axis vector U| is converted
to inertial coordinates by a matrix multiplication as

fix"

i

C

in>

r-f-

1

-

N

U£n

U-

-U£r_

(49)

wiiere N is the matrix that relates tangential, normal, and radial coordinates
to EC1 (x, y, coordinates.

The desired unit vehicle pitch axis is arbitrary from a guidance
standpoint and is selected by the guidance equations to conform to vehicle
and/or telemetry antenna constraints. Finally, the desired unit yaw axis
is formed from U^ x to complete a right-hand orthogonal coordinate
system.

6. MANEUVERING EQUATIONS

This section presents a derivation of the maneuvering equations.

In this paper, maneuvering is considered the application of rate limits to
the guidance commands (desired vehicle axes) and, if the desired maneuver
is large, the computation of an efficient maneuver. In addition, the maneuver-
ing equations calculate values to interpolate between successive major cycle
commands on a minor cycle basis. The attitude error equations, performed
on a minor cycle basis, take the output commands from the maneuvering
equations in the form of desired vehicle axis vectors. The desired vehicle
axes are compared with the actual present vehicle axes (read from the 1MU
attitude sensors), and vehicle attitude errors are computed. The flow is
from guidance equations to maneuvering equations to attitude error equations.

-28-

The initial calculations made by the maneuvering equations are a
coordinate conversion from earth centered inertial (ECI) to drifted launch-
centered inertial (LCI), as follows:

[rg]t = [0]T [*][cg]

u£r - [RG] [U$]

5 6 (50)

U^ = [RG] [Uc]

where U^c, U^, and U^c are the final desired commanded- vehicle roll,
pitch, and yaw axes in drifted LCI coordinates: [CG] is a coordinate trans-

formation matrix from ECI to LCI coordinates; [p] is the present compen-
sable IMU drift matrix; and 4* is a matrix chosen to rearrange the rows of <p
so as to be compatible with the LCI coordinate system.

The drifted LCI coordinate system^ is used throughout the following
equations, since it is the most convenient coordinate system for the guidance
minor cycle calculations; the IMU attitude readouts are referenced to the
drifted LCI coordinate system. The drifted LCI system (called a, b, g) can
also be considered an ideal gimbal angle system.

Following the coordinate transformation, one can calculate the
transformation matrix of the desired maneuver:

u£c '

u4a

U!c-

V

u4c ‘

u r

M =

^nc '

°4a

“V

°na

°nc-

U„

4a

_V '

uu

V'

V

Uy

4a

where Uc , U , and Ur are the present desired commanded- vehicle roll,
pitch, and yaw axes. The trace of the BG matrix can be related to the
commanded maneuver as

-29-

Trace - 1 - 2 cos 8

(52)

o r

cos 0 (1 - Trace)/2

where 0 is the magnitude of the maneuver.

Cos 0 can be tested to determine whether to command a rate-
limited maneuver. If cos 0 is sufficiently close to 1.0, the maneuver is
small and the following calculations are performed:

dU!L = U^c ' UU

and

(53)

ciU T = U - U

ol 0c na

The variables dU^ and dU^j represent the motion of the commanded-body
axes Ut and U during the next major cycle. If dUf. T and dU T are multi-
plied by the reciprocal of the number of minor cycles per major cycle, C^q,
the commanded-body axes can be interpolated on a minor cycle basis. With

6R

= dUt T

cm

oP

= dU T

cm

nL

(54)

computed on a major cycle basis, the following minor cycle calculations
yield the interpolated axes:

-30-

cm

(55)

U|dm

^>ldm

«- u

<- u

^dm

qdm

+ 6 R
+ 6P

cm

If cos 0 is not close to 1.0, a large maneuver has been commanded
and rate- limiting is desired. The maneuvering strategy selected consists
of maneuvering about a single inertially fixed axis at a constant angular
rate. It can be shown (Euler's theorem) that this single-rotation axis is
an eigenvector of the maneuver transformation matrix given in Eq. (51).

The following paragraphs describe an algorithm for computing a command
“eigenvector" maneuver.

To solve for an eigenvector, E, of BG, the maneuver transforma-
tion matrix, consider a skew- symmetric matrix D, calculated as

D = BG - BGT (56)

The D matrix has the properties

[D]

[E]

= to]

(57)

D. .
!J

= 0

if i = j

(58)

From these properties of the D matrix, the following set of
equations is derived:

D12E2 + D13E3 " 0

'D1 2E1 + D23E3 = 0

-°13E1 ' D23E2 " 0

(59)

-31-

Solving for the eigenvector in Eqs. (59) yields

E1 ' D23 BG23 ' BG32

E2 ” "D1 3 " BG3 1 BG1 3 (60)

E3 ^ D1 2 = BG1 2 ' BG21

Alter unitization, the eigenvector is transformed from body
coordinates to the LCI coordinate system with the following matrix
multiplication:

RL = [A]

UE.

UE.

UE

3 J

(61)

where

U

UJ

1 the body axes as columns.

The magnitude of the eigenvector computed in Eq. (60) is propor-
tional to the sine of the maneuver angle 0. There is a singularity wherein
the magnitude of the eigenvector approaches zero when the commanded
maneuver approaches 180 deg. When cos 0 is sufficiently close to -1.0,
the RL vector is replaced by a unit- commanded body axis according to the
following strategy.

The largest diagonal element of the BG matrix is determined.

-32-

Then, if

BG largest, R L. = IL
1 1

BG,_ largest, RL _ U

U L* M <

BG,, largest, lOl = U
33 t,a

The above procedure ensures that the initial maneuver away from
the 180-deg region is not made about an axis which is perpendicular to the
desired maneuver axis.

RL is considered the unit command rotation vector, and the

variables dU^^ and dU^ are computed as

dUf j = (RL X U^a) MLIM

dU = (RLX U ) MLIM
PL ' Pa/

and the interpolated body axes, on a minor cycle basis, are computed as
described in Eqs. (54) and (55). MLIM is the magnitude of the rate limit.

The method used to interpolate the commanded-body vector throughout
the major cycle results in a very small nonorthonormality in the commanded
vehicle axes. To prevent this nonorthonormality from growing, one can per-
form an orthonormalizing process for each major cycle, as follows:

Lu = + dTJei.)/llJu + dLiJ

L,a=1haX %a*«OnL,/|U5aX (U,a + dUnL)l (64)

«na = %X V

This process resets the command- vehicle coordinates to orthonormality
every major cycle.

-33-

c.

NAVIGATION

The- guidance navigation calculations compute the 'present vehicle
position and velocity using as inputs accumulated incremental velocity pulses
from a triad of force rebalance integrating accelerometers. One of the
accelerometers (WC1 is mounted on the IV U stable platform (turret); the
other two (l;C and VC) are mounted on a platform that rotates, or carousels,
at I rpm. Figure 4 >s a block diagram of the major and minor cycle
navigation .

1. ACCEL fill O V 1 1 T E R RESOLUTION AND COMPENSATION

For the navigation function to he performed, the initial azimuth
(at "go inertial") of the rotating platform must be determined. Further,
whenever the incremental velocities are sampled, this azimuth must be
updated.

Four coordinate systems are used in the accelerometer resolution
and compensation:

[uc, vc, wcj

[l,2,3]

[u, v,w]

[x,y, z]

Actual ca rouselling accelerometer input axes
coordinate system. The coordinate system is, in
general, nonorthogonal and rotating due to 1MU drift
and ca rouselling.

Ideal carouselling coordinate system This coordi_-
nate system of convenience is defined as follows. 3
is colinear with the carouselling axis of rotation.

2 is perpendicular to 3; it is in_the plane of 3 and the
vc accelerometer input axis. 1 completes the
orthogonal right-hand set.

Drifted launch site coordinate system. Initially
equal to the initial vehicle roll, pitch, yaw coordi-
nate system. After go inertial, related to the initial
vehicle coordinate system by the IMU drift matrix.

Earth centered inertial coordinate system. Defined
as follows:

x = unit vector in the equatorial and prime meridian
planes at go-inertial time.

z = unit earth spin vector, i.e., North Pole

y - z x x to complete a right-hand set.

34

Figure 4. Navigation Block Diagram

An miii.il >1/. 1 ‘nuih, or phase angle, is computed in the ground-in-
flight interface program as follows:

FANG - HTHYN + CFANG + CZZANG

(65)

where

HTHYN = the angle, computed in the ground alignment program, from
north to the VC accelerometer input axis plus gyro mis-
alignment errors

CZZANG = gyro misalignment IMU compensation parameter

CFANG = parameter to convert the phase angle from northeast to

dov/nrange-crossrange coordinates and to time synchronize
with the accelerometer readouts.

The sine and cosine of the initial phase angle are then taken for resolution
purposes .

SS = sin(FANG)
BS = cos (FANG)

(66)

After initialization, the carousel phase angle sine and cosine is
updated every 40 msec by a trigonometric identity as follows:

SSn+l = <SSnHcos 0.24°) + (BS )(sin 0.24°)
BSn+1 = (BSn)(cos 0.24°) - (SSJisin 0.24°)

(67)

where 0.24 deg is the angle :’car luselled" through in 40 msec. Also, the
carousel phase angle is reinitiaJ zed to SS and BS at 1 -minute intervals.

The accelerometer incremental counts are modified with the calibra-
tion constants determined as part of factory calibration and pad test procedure
The 40-msec frequency accelerometer compensation equations are

AVuc = (CKAX)(AN - CKBXG) + (CKNX)(AN )2

uc

A ^vc (CKAY)(ANvc - CKBYG) + (CKNY)(ANv(J2

AVwc = (HZAF)(AN - HBZ) + (CKNZ)(AN )2

wc

whe re

ANuc’ ANvc’ ANwc the raw counts from the carouselling uc and vc

and the stationary wc accelerometers over the
last 40-msec interval

CKBXG, CKBYG = the uc and vc accelerometer factory bias

calibration

HBZ - ^e wc accelerometer bias calibration computed
during final align

CKAX, CKAY = the uc and vc factory scale factor calibration

HZAF = the wc accelerometer scale factor calibration
computed during final align

CKNX, CKNY, CKNZ = the uc, vc,

and wc nonlinearity calibration

Continuing, one finds that

ST1 = AVuc " (czzb2U)(AV ) - (CZZB3U)(A V t

vc wc’

ST2 = AV - (CZZB3V)(AV )

VC WC'

(69)

Equation (69) shows small angle approximations to misalignment
rotation compensation where

ST1 , ST2 = velocity increments in 1,1,3 coordinates

CZZB2U - misalignment of the u axis toward ~Z

CZZB3U = misalignment of the u axis toward 3

CZZB3V = misalignment of tne v axis toward 3

-37-

ST1 and ST2 are then resolved into u,v,w coordinates as follows:

IBl

.

mm

AVum = (BSHST1) - (SS)(ST2)

AVvm = (SS)(ST1) + (BS)(ST2)

The resolved minor cycle velocity increments are finally
accumulated as follows:

A V <- AV A V
up up um

AV AV + AV
vp vp vm

[T

J I

■ *>W'! *

A V f AV + A V
wap wap wc

Once per second, the accumulated velocity increments are sampled
for use as inputs to the major cycle navigation equations

AV

«- AV

u

up

AV

*- AV

V

vp

AV

«- AV

wa

wap

and AV

are chei

up vp wap

further accumulation in the next major cycle interval.

AV 0
up

AV «- 0
vp

AV <- C
wap

The w-accelerometer misalignments are performed on a major
cycle basis, since the w instruments do not carousel. These misalignments
do rotate, because the w-accelerometer is rotating at earth rate at liftoff.
During guidance initialization, misalignments are computed in the u, v, w
coordinate system as a function of the initial gimbal angle

bWl = (CKB4W)(sin aj + (CKB5W)(cos a)

BW2 = - (CKB4W)(cos a.) + (CKB5W)(sin a.)

where

BWl - w-accelerometer misalignment away from u axis

BW2 - w-accelercmeter misalignment away from v axis

CKB4W, CKB5W = factory-calibrated w-accelerometer misalignments

in an a gimbal-oriented coordinate system

The w-accelerometer input axis also "cones" after go inertial as a
function of the inner gimbal angle, because of misalignment of the inner
gimbal axis and the carouselling axis. The complete equation for
w-accelerometer misalignment compensation is given in Eq. (75):

AV = A V + [BWl + (CKBSW)(cos a) - (CKBCW)(sin a)]AV
w w & u.

+ [BW2 - (CKBCWMcos a) - (CKBSW)(sin »)]AV

CKBSW, CKBCW = calibrated carouselling axis misalignments in an

a gimbal angle coordinate system

-39-

2.

GYRO DRIFT COMPENSATION

Gyro drift compensation is accomplished by computing the platform
drift rate with calibration values obtained during hangar and pad tests. Then
by time -integration of the drift rate, a matrix p can be computed that trans-
forms the present (u,v,w) axes to the initial nondrifting axes. Further
matrix operations then transform to the desired ECI coordinate system.

rAV l

rAV -i

sx

u

AV

sy

= [CG]r[4>]T[p]

AV

V

AV

L szJ

AV

L w_

(76)

where

AV

sx, sy, sz

measured (sensed) values of velocity increments in the
ECI coordinate system

AV

u, v, w

measured value of velocity increments along the, present
drifted launch site coordinate system

p - d rift matrix

4* row rearrangement matrix

CG - coordinate transformation matrix from LCI to ECI
coordinates

Note that p, cp and CG are shared by guidance (Paragraph III.B.6) and
navigation.

If the instantaneous drift rate vector is given as (pu> pv, pw), the
derivative of each column of the p matrix is the cross-product of the drift
rate vector with that column vector. Calculating the cross products and
combining terms, one obtains the matrix form

p = [p]

0

6

\

w

-p

0

< P ,

w

-p

u

0

(77)

Integration of p is accomplished by using a second order Runge-
Kutta algorithm. First, the p matrix is integrated a half cycle with an
initial derivative:

K~~ *n + <At/2)*n (78)

The drift rate is then averaged over two cycles:

A p = (A p + A p , , ) / 2
ru run run+l

A p = (A p + Ap , . ) / 2 ( < ^)

rv rvn rvn+l'

A? = (A p + A 0 , , )/2

Tw rwn rwn+l'

A final step advances p as follows:

n+1

1 Pn +

[p 1
L Vn ,

0

A ^

-Ap

rw

Ap

rw

0

-A?u

-Ap

rv

Ap
y u

0

(80)

Computation of the drift vector ip in u, v,w coordinates is complicated
by the rotation of the carouselling instruments. Drifts of each gyro must be
located spatially due to the arbitrary location of the 1MU turret and the
rotation of the platform at 1 rpm.

All drift matrix calculations are performed once per second. A
central assumption for the drift calculations is that the drift occurred approxi-
mately centered in the 1 sec compute cycle. To locate the carouselling
instruments, one must compute backed-up values of the variables BS and SS.

SSB = -(BS)(sin 3°) + (SS)(cos 3°)

BSD - (BS)(cos 3°) + (SS){sm 3°)

(81)

The variables SSB and BSB are then used to resolve AV and AV into

u v

coordinates representing the "average" position of the carouselling gyros
over the last second.

AVgi = (AVu)(BSB) + (AVv)(SSB)

AVg2 = -(A~Vu)(SSB) + (AVv)(BSB)

(82)

The fixed-torque and unbalance drifts of the carouselling gyros are then
computed as

DRFTU FT DU + (CKU1)(AV t) + (CKU2)(AVg2)

DRFTV = FTDV + (CKU3)(AVg2) + (CKU4)(AVgJ)

(83)

where

FTDU, V = fixed-torque drifts
CKU 1 , 3 = spin-axis unbalance drifts
CKU2, 4 = input-axis unbalance drifts

In addition to the usual error sources associated with an IMU,
carouselling itself introduces gyro drifts. The compensable ones are called
gimbal- oriented bias (GOB), turret-oriented bias (TOB), and turret-oriented
eta (TOE). GOB is a drift in the carouselling plane, which rotates as the inner
gimbal rotates; i. e. , a drift fixed to the inner gimbal. Initially, GOB is
computed in launch-centered coordinates as

-42-

GOBC1 = (CK20)(cos CK21)
GCBC2 = (CK20)(sin CK2.1)

(84)

where

CK20 - GOB magnitude
CK21 = GOB phase angle

7

Later, in flight, the computation of GOB reflects the rotation about
the inner gimbal as follows:

GOBI - - (GOB C2)(cos a) + (GOB Cl)(sin a)
GOB2 = (GOB Cl )(cos a) + (GOB C2)(sin a)

(85)

I urret- or tented bias is a drift of the tui ret (perpendicular to the
carouselling plane) that is a function of several harmonics of the turret present
position with respect to the inner gimbal. To locate the turret during
guidance initialization, one must compute the angle of the turret revolution
from a zero inner gimbal angle value:

- tan (sin Q-. /cos a. )

where

°'1 - initial (at go inertial) inner gimbal angle
sin a. = sine of initial a
cos or. = cosine of initial a

-43-

Then, in flight, this a,ngle is updated as

- tan (sin a! cos a) +

(87)

The equation for TOB is

4

TOB = [cos (n a - 0 )

f— { iL 'it l

i=i

(88)

In Eq. (88), n. represents integers specifying the harmonics of a ^ to
be included in the TOB calculation,, 6^ represents the phase angle of each
harmonic; and K represents magnitudes.

TOE is a drift in the carouselling plane that is a function of the
location of the turret with respect to the launch- centered coordinate system.
It is computed at guidance initialization as

TOE 1 = - (CK22)[ (sin a.)(cos CK23) + (cos *.)(sin CK23) J
TOE2 = (CK22)l(cos m)(cos CK23) - (sin a.)(sin CK23) ]

(89)

who re

CK22 TOE bias drift magnitude
CK23 TOE bias drift phase angle

Finally, the derivative of <p is computed as shown in Eq. (90):

A pu = (BSBKDRFTU) - (SSB)(DRFTV) + GOBI + TOE1
Apv = (SSBMDRFTU) + (BSB)(DRFTV) + GOB2 + TOE2 (90)

A pw = HRZ + (DIAU)(AV ) - (CZZU6)[ (AVv)(sin a.) + (AVJfcos ».)] + TOB

-44-

V

whe "e

3.

BSB, SSB = sine and cosine of the average carousel angle
during last compute cycle

DRFTU, DRFTV = fixed-torque and g-sensitive drifts of the UC and

VC carouselling gyros

GOBI, GOB 2 = GOB drift [Eq. (84)]

TOE 1 , TOE 2 = TOE drift [Eq. (89)]

TOB = TOB drift [Eq. (88)]

HRZ = noncarouselling Z gyro fixed-torque drift

DIAU - spin-axis unbalance drift

CZZU6 = input-axis unbalance drift, Z gyro

TRANSFORMATION OF VELOCITY TO INERTIAL x,y,z
COORDINATE SYSTEM

The inertial x, y,z coordinate system is defined as follows:

x = unit vector in the equatorial and prime meridian planes at
go inertial time

z = unit earth spin vector, i. e. , North Pole
y = z X x to complete a right-hand set

4. VEHICLE INERTIAL POSITION

The following trapezoidal integration formulas are used to obtain
the vehicle position (X, Y, Z) in the inertial coordinate system:

X <■ X + V (At) + (1/2) (At)2 (AV + AV )

Y «- Y + V (At) + (1/2) (At)2 (AV + AV ) (9D

y y o y

Z Z + V (At) + (1/2) (At)2 (AV + AV )

Z b Z gZ

-45.

where

V , V , V
x y z

At

AV , A V
sx

AV , A V

gx

sy

gy

» A V

sz

,AV

gz

x, y, z components of vehicle velocity

major compute cycle interval - 1 second

x,y,z components of sensed vehicle velocity
increment from Eq. (76)

x, y, z components of velocity increments due to
gravity

Vehicle position magnitude is then calculated as

2 2 2 1/2

r = (x + y + z )

(92)

5. GRAVITY COMPUTATIONS

An approximation for the gravitational potential of the earth is

U

U = (n/R)[l + (Ja2/3R2)(1 - 3 sin2\) ] (93)

where

p = gravitational parameter of the earth
R = distance from center of the earth
\ - latitude

a = equatorial radius of the earth

J = a characteristic constant that is a function of the moments of
inertia with respect to the polar and equatorial axes

A derivation of Eq. (93) is contained in Ref. 7. This approximation,
which is a truncated series, contributes negligible inaccuracies to the overall
navigation function.

-46-

Then, from the definition of the Z component of vehicle position,
it follows that

sin2 \ = 22/R2 (94)

which, upon substitution into Eq. (93), yields

U = (p/R) (1 + Ja?'/3R2 - Ja2Z2/R4) (95)

The x, y, z components of the acceleration due to gravity are then
obtained by the partial differentiation of Eq. (95) with respect to each axis,
and

gx = -aU/9x = -(8U/9R)OR/ax)

gy = -au/ay = (-au/aR)OR/ay) (96)

g = -au/az = (-au/aR)OR/az)

The negative sign is added by convention.

An evaluation of the common term 3U/aR yields

aU/aR = (-p/R2)(l + Ja2/R2 - 5Ja2Z2/R4) (97)

2

and, by factoring out - -(p/R ), one obtains

aU/3R = A (1 - Ja2A /p + 5 Jr.2 A Z2/pR2) (98)

8 8 8

-47-

In addition, one finds that

dR/dx = X/R
9R/9y = Y/R
dR/dx = Z/R

(99)

and

aU/az = ZfxJai 2Z/R5 = +2JaZA2Z/|j.R

8

(100)

The velocity increments over a major compute cycle due to gravi-
tational acceleration arc given in Eq. (101).

AVgx gx -(ou/aR)(aR/ax)

a v g - -(au/aR)(aR/ax)

Bi y

AV =g - -OU/9R)OR/8z)

gz bz

6. VEHICLE INERTIAL VELOCITY

The inertial velocity of the vehicle is computed as

(101)

(V

.). (V) + (1 /2)[(AV .) , + (AV .) ] [At] + (AV )(At)

in) 1 in L gi n+1 gi nJ 1 J si

i = x,y, z

gi

(102)

Averaging AV over two cycles produces a trapezoidal integration of acceler-

D. ATTITUDE ERRORS

Attitude errors art? < rucial outputs of the guidance equations utilized
by the Digital Flight Controls Equation in stabilizing the vehicle to a desired
attitude. They represent differences between the present actual vehicle
attitude and the present desired vehicle attitude. The desired attitude is
computed by the guidance maneuvering equations (Paragraph III. B. 6); the
actual attitude is derived from synchros attached to each gimbal.

1 .

form

SYNCHROS

The synchro signals supplied to the inflight computer are of the

VI = K sin (0-120°) and V2 = K sin (0-60°)

(103)

where G can be any of three gimbal angles, a, p, or y„, and K is a scaling
factor common to VI and V2.

By trigonometric identities, one obtains

VI - V2 = K sin (0-120°) - K sin (8-60°)

= K[(sin 0 cos 120° - cos 0 sin 120°)
- (sin 6 cos 60° - cos 0 sin 60°)]

= K sin 0 (cos 120° - cos 60°)

(104)

-K sin 8

-49-

a n<l

VI I V2 K sin (0 - 120°) + K sin (0 - 60°)

K | (sin 0 cos 120° - cos 0 sin 120°)

+ (sin 0 cos 60° - cos 0 sin 60°) |

(105)

- K cos 0 (- sin 120° - sin 60°)

= -K*/3 cos 0

Thus, by simple sum and difference, one obtains the sine and
cosine of the gimbal angles, except for a common multiplier -K and the
constant V 3.

Since the sine and cosine of the gimbal angles are desired on a
minor cycle basis as a convenience for the computation of vehicle attitude
errors, it is necessary to determine the scaling factor K.

At go inertial, the following initial calibration is performed for
each of the three gimbal angles:

X - V2 - VI
Y = (VI + V2)(C1)

(106)

K2 - (X)(X) + ( Y)( Y)

K -v/ K2

In this series of equations. Cl is a constant equal to -1.0/\f3, and K
is the calibrated value of the synchro scale factor.

On succeeding minor cycles, the sine and cosine of each gimbal
angle are computed as follows:

-50-

X = V2 - VI
m

Ym = <V1 + V2>(C1)

K2 - (X )(X ) + (Y )(Y )

m m m m m

K - (1/2) fK2 /K + K )
m m m rn

107)

sin 0 = X/K
cos 0 = Y/K

m

m

These minor cycle computations use Newton’s square-root algorithms,
which converge quite rapidly if the initial value is relatively accurate.

Since the three gimbals, n, (3, and y^, are not necessarily zero at
go inertial, they are read initially; the initial values are used to compute the
three angles through which the vehicle rotates during the flight. Taking a as
an example, trigonometric identities yield

sin a = sm a cos a. - cos a sin a
pi pi

cos a = cos a cos a. + sin a sin a
pi pi

(108)

where

sin a, cos a = sine and cosine of vehicle angular rotation
sin a , cos a = present gimbal synchro sine and cosine readings
sin o , cos m = initial gimbal synchro sine and cosine readings

An identical procedure is followed to compute the sine and cosine of the }3
and vD vehicle rotations.

-51-

The fourth gimbal y has very limited travel (‘"10 deg), and the
following simplifying approximations are made:

cos y - 1.0

(109)

sin y ~ (CKDSCA}(sin 'y ) - sin y^

where CKDSCA = calibrated scale factor for sin

ATTITUDE ERROR COMPUTATIONS

The IMU gimbal angles (a, p, Y and Y^) ideally establish the
relationship between two vehicle coordinate systems: the present roll,
pitch, and yaw (£,q,p body axes; and the initial body axes (£.,C,£.l. Initial
alignment is such that, at launch, a positive roll results in a negative gimbal
angle «; a positive yaw results in a positive gimbal angle (3; and a positive
pit< h results in a positive gimbal angle Y^. The fourth gimbal Y, in normal
circumstances, is zeroed. Its sign convention is the same as Y

i\.

[uc.r,|.u,

CO* tp 0 -sin

0 I 0

\K <» * o« \

(' sin P o i » ■<*> >

i

- sin ,1 ( ,1 1(1 (j

I I

ljr, 0

.] (110)

ij si n it i 06

I 06 / |

Performing the matrix multiplication with cos y ~ 1.0 yields

-52-

(' '>■> 1 1(, ‘ " i“i ) l( --'ii 'i )

(» ■ >& \ ^ sin 6 cos u

( - cos ^ ^ sin >i sin /

- cos cos 0 sin \ sin a

- cos y^ c ob 3 sm \ cos a

- -nn ^ sin o)

- sin Yp cos <>)

(- bin |i)

(t os 3 cos o +

( - cos (1 sin a + sin 3 sin \ tos

sir* 3 sm y s in a

(sin > ^ cos il t cos y ^ sin y

(sin Y{^ sin 3 cos a

(- sin y ^ sin 3 sin u

- sin cos p sin y sin o

- 6in y^ cos 3 sin y cos a

+ cos sin »)

f cos y . Cos o)

:n i)

or M = lUg, U^, U^j

If the matrix of initial vehicle axes is assumed to be identity, the
rows of the M matrix comprise the present measured vehicle axes in a
gimbal-oriented coordinate system (a,b,g), where initially

a = Uc

b = U

ni

(112)

From the actual vehicle axes and the present desired vehicle axes, attitude
errors can now be computed.

Consider a three-axis (roll, pitch, yaw) maneuver to reorient the
vehicle from its actual attitude to the desired attitude, as three Euler
rotations in the following order: roll, pitch, yaw. In matrix form, tins is

-53-

0

0

B O cos R sin R

cos P 0 -sin P I f cos Y sin Y 0 ~J

-sin Y cos Y 0

0 - sin R cos R sin P 0 cos P

where R, P, and Y are the respective roll, pitch, and yaw rotations.
The matrix multiplication yields

(113)

cos P cos Y

-cos P sm Y

sin P

sm R sin P cos Y
+ cos R sm Y

sm R sin P sin Y
+ cos R cos Y

-sin R cos P

•cos R sin P cos Y
+ sm R sin Y

cos R sin P sin Y
+ sm R cos Y

cos R cos P

(114)

Simplifying B with small angle approximations yields

"1 Y -P

B« -Y 1 R
P -R 1

(115)

Another expression for B in terms of body axes is

§dm

-r,5

^ £dm

• U

n

Ut .

§dm

•U£

>

B -

0 .
pdri)

'°£

0 .
i|dm

• U

n

U .
r)dm

(116)

%m

f,dm

• U

n

i

t,dm

•DJ

REFERENCES

1 E. Da maud , D. L. Kleinbub, and J. B. Shaul, Guidance, Control
and Ground Equations for SSLV C-26, Aerospace Corporation Report
No. TOR- 01 7 2(21 { 2- 02)- 1 1 , Reissue B (1 November 1972), Vol. I.

Program 624A Mission Specification for Flight Plan VII- J,

Aerospace Corporation Report No. TOR-01 72(21 1 2- 02)-6 , Rev. 1
(28 September 1972).

H. Sokoloff, Program 624A Discrete List for Flight Plan VII- J,
Aerospace Corporation Report No. TOR-0172(21 12-02)-l6,

Reissue A (24 November 1972).

George W. Cherry, "A General Explicit Optimizing Guidance Law
for Rocket Propelled Spacecraft, 11 Proceedings of the AIAA/ION
Astrodynamics Guidance and Control Conference, AIAA Paper
No. 64-638 (24-26 August 1964).

D. MacPhcrson, An Explicit Solution to the Powered Flight
Dynamics of a Rocket Vehicle, Aerospace Corporation Report No.
TOR- 1 69(31 26)TN-2 (31 October 1962).

C. W. Pittman, The Design of Explicit Guidance Equations for
Rocket Ascent, Aerospace Corportation Report No.

T DR- 469(5 5 40 - 1 0)- 4 (24 May 1965).

F.E. Darnaud, A Technique Allowing Continuous Operation of
Integral Control, Aerospace Corporation Report No.

TO R- 1 00 1 (2 1 1 6 -60)- 8 (1 3 September 1966).

K. A. Ehricke, "Environmental and Celestial Mechanics,"

Principles of Guided Missile Design, Vol. I: Space Flight
(D. Van Nostrand Company, Inc., Princeton, N. J. , I960).

Carousel V Inertial Navigation System, System Technical Description
EP0137, Delco Electronics Division of General Motors, Milwaukee,
Wisconsin, 1969.

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